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The Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations via Graph Partitioning

Shriram Srinivasan, Kaarthik Sundar

TL;DR

This work tackles the challenge of solving potential-driven, steady-state nonlinear network-flow equations on large graphs by partitioning the network into tractable subdomains and solving a global problem through local, smaller subsystems. A bi-level reformulation is introduced, where interface and non-interface nodes are treated with slack variables, enabling an outer Newton step on a reduced system that relies on an inner solve of subnetwork systems; this approach is shown to be equivalent to a Schur-complement update of the full system. A general network partitioning approach using permissible vertex separators is developed to produce subnetworks that can be solved in parallel, with data sharing constrained to interconnects between subnetworks. The method is demonstrated on GasLib test networks, including a Texas network partitioned into 9 subnetworks, yielding accurate results and validating the practicality of the partitioned, privacy-preserving strategy for large-scale network flow problems.

Abstract

The solution of potential-driven steady-state flow in large networks is required in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and its solution grows more challenging as the network size increases. We present an algorithm that utilizes a given partition of a network into tractable sizes to compute a global solution for the full nonlinear system through local solution of smaller subsystems induced by the partitions. When the partitions are induced by interconnects or transfer points corresponding to networks owned by different operators, the method ensures data is shared solely at the interconnects, leaving network operators free to solve the network flow system corresponding to their own domain in any manner of their choosing. The proposed method is shown to be connected to the Schur complement and the method's viability demonstrated on some challenging test cases.

The Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations via Graph Partitioning

TL;DR

This work tackles the challenge of solving potential-driven, steady-state nonlinear network-flow equations on large graphs by partitioning the network into tractable subdomains and solving a global problem through local, smaller subsystems. A bi-level reformulation is introduced, where interface and non-interface nodes are treated with slack variables, enabling an outer Newton step on a reduced system that relies on an inner solve of subnetwork systems; this approach is shown to be equivalent to a Schur-complement update of the full system. A general network partitioning approach using permissible vertex separators is developed to produce subnetworks that can be solved in parallel, with data sharing constrained to interconnects between subnetworks. The method is demonstrated on GasLib test networks, including a Texas network partitioned into 9 subnetworks, yielding accurate results and validating the practicality of the partitioned, privacy-preserving strategy for large-scale network flow problems.

Abstract

The solution of potential-driven steady-state flow in large networks is required in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and its solution grows more challenging as the network size increases. We present an algorithm that utilizes a given partition of a network into tractable sizes to compute a global solution for the full nonlinear system through local solution of smaller subsystems induced by the partitions. When the partitions are induced by interconnects or transfer points corresponding to networks owned by different operators, the method ensures data is shared solely at the interconnects, leaving network operators free to solve the network flow system corresponding to their own domain in any manner of their choosing. The proposed method is shown to be connected to the Schur complement and the method's viability demonstrated on some challenging test cases.
Paper Structure (10 sections, 2 theorems, 16 equations, 2 figures, 1 algorithm)

This paper contains 10 sections, 2 theorems, 16 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Governing equations
  3. A bi-level reformulation
  4. Relation to Schur complement
  5. Network partitioning with the bi-level formulation
  6. Intuitive description
  7. Mathematical description
  8. Observations and Comments
  9. Results
  10. Conclusion

Key Result

Lemma 1

For a connected graph $G$, a permissible vertex separator $\mathcal{C}_V \subset V(G)$ can be used to construct $N(\mathcal{C}_V) \geqslant 2$ subnetworks of $G$, $S_1, S_2, \dotsc S_{N(\mathcal{C}_V)}$ such that

Figures (2)

  • Figure 1: The GasLib-40 network is shown with interface nodes coloured black, while $S_1, S_2$ are shown in different colors with the interface nodes belonging to both. With the interface nodes as slack, can solve two subsystems corresponding to $S_1$ and $S_2$ respectively.
  • Figure 2: Partitioning of Texas network with 2451 vertices birchfield2024structural into 9 subnetworks, all of size less than 500 vertices. The 25 interface nodes are coloured black.

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Definition 1: Bi-level formulation
  • Remark 3
  • Definition 2
  • Definition 3: Vertex Separator
  • Definition 4: Permissible vertex separator
  • Remark 4
  • Definition 5: Vertex removal
  • Lemma 1
  • ...and 6 more